Solve each rational inequality. Give the solution set in inter...

Question

Solve each rational inequality. Give the solution set in interval notation.

$\frac{5}{2 - x} > \frac{3}{3 - x}$

Accepted Answer

Provide the solution set for the given rational inequality and interval notation where we have one divided by 13 minus X is greater than two divided by 17 minus X. Now, to solve this inequality, you want this inequality to be greater than zero. So I'll move my two divided by 17 -1 to the left side. You will then get one divided by 13 minus X minus two divided by 17 minus X square than zero. Now, to combine this, we all have to have the same denominator on both fractions. Let's multiply the left fraction by 17 minus X divided by 17 minus X. Same for the other one. We will multiply instead by 13 minus X divided by 13 minus X. We will then get 17 -1. Goodbye. Goodbye. 17 minus XX multiplied by 13 minus X minus two multiplied by 13 - divided by 17 minus X multiplied by 13 minus X, all of that square than zero. Now, we can combine 17 minus X minus two times 13. It's 26 and two multiplied by negative picks with a negative gives us plus two X all this divided by 17 minus X 13 minus X. We can then simplify again to gets X minus nine divided by 17 minus X multiplied by 13 minus X, all of that greater than zero. Now, we can find our discontinuities and intercepts. Let's set 17 minus X equals to 0 13 minus X equals zero and X minus nine is equal to zero. You will then get X is equals to 17, X is equals to 13 and X is equals to nine. We now have all of our intervals. Our intervals will be from negative infinity to none, 9 to 13, 13 to 17 and 17 to Infinity. Now, we can go ahead and check our intervals for anything greater than zero to do that. We will take values on every one of our intervals and plug it into our equations. Let's see the first one which is 8 f of 8, then will be 8 -9 divided by 17 -8, multiplied by 13 -8. That would give us native one divided by 45. Let's check another value F of 10 on our next interval with the sub, the substitute are 10. It's every value we end up getting one divided by 21. That's another value 14. If I were to plug that in, I would get negative five divided by three. Finally, let's test one more value on our final interval. So we use 18, we plug that into our equation. We would get nine divided by five So we notice here there are two intervals that have positive numbers that was being 9 to 13 and 17 to Infinity. We can then determine the solution set to our problem corresponds to answer B OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Solve each rational inequality. Give the solution set in interval notation.
$\frac{5}{2 - x} > \frac{3}{3 - x}$

Key Concepts

Rational Inequalities

Finding a Common Denominator and Cross-Multiplication

Domain Restrictions and Interval Notation

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